In an exercise I've proven that $\partial(\partial A) \subset \partial A$, for any $A\subset X$, where $X$ is a topological space and $\partial$ in this case stands for the boundary. Apparently, in general the equality does not hold, but I'm stumped when it comes to thinking of a counterexample. My problem is that for all of the spaces and sets I think of, the boundary always ends up being closed and with empty interior, so when taking the boundary a second time, I end up with the same.
Could someone come up with such an example please? Thank you.
Hint. Consider $(0,1)\cap\mathbb{Q}$ in $\Bbb{R}$ with usual topology.